Lüli rongtong 律曆融通

Lüli rongtong 律曆融通 "The Harmonization of pitch and calendar" is a book on astronomical calculations in combination with the calculation of musical pitches. The book of 4 juan length is part of Zhu Zaiyu's 朱載堉 (1536-c. 1610) collection Yuelü quanshu 樂律全書.

Zhu Zaiyu's œuvre of 23 books encompasses a vast array of subjects, including music theory, musical instruments, notation, dance choreography, mathematics, calendar studies, and measurement systems. His work focuses particularly on pitch systems (lü 律), exploring numerical ratios, pitch temperament, and their connection to calendars. His most remarkable contribution was the creation of the twelve-tone equal temperament (shi'er pingjun lü 十二平均律), which was an unprecedented achievement in China and a groundbreaking discovery worldwide.

Zhu Zaiyu's contributions to astronomy and calendrical studies are mainly evident in his work Lüli rongtong. During the Ming period 明 (1368-1644), the officially adopted calendar was the Datong Calendar 大統曆, created by Liu Ji 劉基 (1311-1375), which itself was a modified version of Guo Shoujing's 郭守敬 (1231-1316) Shoushi Calendar 授時曆 from the Yuan period 元 (1279-1368). A key modification in the Datong Calendar was the removal of the correction for the long-term variation in the length of the tropical year (sui shi 歲實).

Although the Shoushi Calendar was precise for its era, its inaccuracies became more evident over time. The elimination of the precession correction (suishi xiaochang 歲實消長) in the Datong Calendar exacerbated these errors. For instance, by Zhu Zaiyu's time, the predicted date of the sun's perihelion (taiyang zuisu dian 太陽最速點) had already shifted by six days, and mistakes in predicting solar and lunar eclipses had become commonplace. A notable incident occurred in 1592, when the Directorate of Astronomy (qiantianjian 欽天監) miscalculated a lunar eclipse by a full day. In 1595, Zhu Zaiyu and the astronomer Xing Yunlu 邢雲路 (d. 1549; author of Wushen lichun kaozheng 戊申立春考證) submitted separate petitions advocating for calendar reform and presented their own revised calendar systems.

Zhu Zaiyu submitted several works to the throne, namely Lüli rongtong, Shengshou wannian li 聖壽萬年曆, and Wannian li beikao 萬年曆備考, which contained an examination of past winter solstices in calendars (Zhuli dongzhi kao 諸曆冬至考), studies on gnomon lengths during the solstices (Erzhi guiying kao 二至晷影考), and a study on historical eclipses (Gujin jiaoshi kao 古今交食考). However, neither Zhu Zaiyu nor Xing Yunlu's proposed calendars were adopted.

The Lüli rongtong comprises the proposed Yellow Bell Calendar (Huangzhong li 黄鐘曆; "Yellow Bell" refers to the basic pitch pipe, a standard measure of length), two fascicles discussing the Huangzhong Calendar (Huangzhong liyi 黄鐘曆議), and an appendix on the interpretation of terms (Yinyi 音義).

The chapter on the Huangzhong Calendar begins by discussing the relationship between musical pitches (lü) and the calendar (li 曆), asserting that the musical pitch serves as the foundation of the calendar, while the calendar is derived from musical pitches. Zhu Zaiyu based this theory on the methods employed by Deng Ping 鄧平 and Luo Xiahong 落下閎 (156-87 BCE) in the Former Han period when they created the Taichu Calendar 太初曆, which incorporated the relations of pitch pipes (huangzhong lü shu 黄鐘曆數). The text explores the calculation of musical temperament (Bu lülü 步律吕), lunar cycles (Bu falian 步發歛), new moons and intercalary months (Bu shuorun 步朔閏), the sun's orbit (Bu richan 步日躔), and sundial shadows and water clocks (Bu guilou 步晷漏).

The fascicles titled Huangzhong liyi elucidate the calculation of lunar distances (Bu yueli 步月離), the lunar nodes (Bu jiaodao 步交道), eclipses (Bu jiaoshi 步交食), and the positions of the Five Planets (Bu wuwei 步五緯). Zhu Zaiyu's Huangzhong Calendar considers the year 1581 as its epoch (li yuan 曆元), but in reality, it is founded upon the preceding "lü-limit year" (lüxian nianfen 律限年份; 1281) as its starting point. The length of the tropical year is computed as 365.241975 days. This results in a deviation of only 21 seconds from the modern theoretical value, rendering it slightly more accurate than the Shoushi Calendar, which has an error of 23 seconds.

With twelve chapters (Lüyuan 律元, Lümu 律母, Lüyi 律義, Lüshu 律數, etc.), Zhu Zaiyu devoted considerable effort to exploring the connections between music theory and astronomy. While some of his discussions offer intriguing insights, the overall argument is not entirely convincing. The most valuable part of this section lies in the chapter Lüyuan, where Zhu critiques the traditional method of calculating the "supreme epoch" (shangyuan 上元) and instead advocates for the method of "cutting off the epochs" (jieyuan shu 截元術). This argument is regarded as a notable and insightful contribution to calendrical reform.

Fascicle 4, comprising twenty-four chapters, is far more practical and concrete than the discussions in Juan 3. This treatise presents suggestions for the precise calculation of solar and lunar eclipses (Rishi 日食, Yueshi 月食), proposes a novel method for measuring the altitude of the Polestar (i.e., geographic altitude; Houji 候極), provides updated measurements of magnetic declination (Zhengfang 正方), and offers explanations on lunar phase changes and eclipses (Jiaohui 交會), demonstrating an approach akin to that of Zhao Youqin 趙友欽 (1271-1335), renowned for his eclipse studies.

Other topics of the fascicle include the need to harmonise the five cycles (Wuji 五紀) of the year, months, days, planets, and calendar; historical changes in the beginning of the year during the "Three Epochs" (Sanzheng 三正, i.e., the beginnings of the years in the calendars of the Xia, Shang, and Zhou dynasties); corrections for inaccuracies in the Shoushi and Datong calendars (Ertong 二統); the reduction of the surplus fraction (Suiyu 歲餘) of the tropical year (determined to 1/2419 of a year); the new moon surplus (Shuoyu 朔餘) established in the Linde 麟德曆 (665) and Jiyuan 紀元曆 (1107) calendars; variations in the lunar cycle and intercalary months (Yingxu 盈虚); the association of hexagrams (see Yijing 易經) with calendar dates (Yaoxiang 爻象); theories on solar motion (Richan 日躔) by He Chengtian 何承天 (370-447) and Yixing 一行 (683-727); amendments of data on the celestial sphere (Tianzhou 天周) at 365°25'75"; precession of the equinoxes (Suicha 歲差); the "longitudes of the planets" (Mingdu 命度); emphasis on the importance of the gnomon shadow (Guiying 晷景), water clocks (Louke 漏刻) and timekeeping (Gengdian 更點); calculations of lunar motion (Yuedu 月度); the effects of solar and lunar velocities on the determination of new moons (Dingshuo 定朔); the orbital inclination of the moon (Jiaodao 交道); determination of the fixed numerical rules according to which the sun and moon move (Dingshu 定數); and data on the motion of the five visible planets (Wuwei 五緯) with direct, retrograde, stationary, and hidden phases (shun 順, ni 逆, liu 留, fu 伏). Overall, juan 3-4 of Zhu's book provide meticulous refinements to previous Chinese calendrical calculations, integrating astronomical observations, and introducing new measurement techniques that improve the precision of Chinese astronomy.

The Lüli rongtong preserves invaluable historical astronomical records that would otherwise have been lost. For example, the Yueshi chapter documents the ancient belief that "the Earth is also round [like the heavens], and not flat" (diti yi yuan er bu fang 地體亦圓而不方). This record is significant, as it proves that ancient Chinese scholars had already recognised the Earth's spherical shape.

In later years, Zhu Zaiyu expanded his research on eclipse predictions, incorporating his findings into the Wannian li beikao. However, he formally presented his specific calendrical revisions in his later work, Shengshou wannian li.